A while back in Nature a paper was published on the most distant confirmed galaxy discovered so far. The paper is behind a paywall, but you can see the arxiv version here. The galaxy, known as z8 GND 5296 has a measured redshift of 7.51. You can see the galaxy in the image above.
So just how far away is this galaxy? It depends on which distance you are talking about.
When determining the distance of far galaxies like this one, astronomers typically give the value purely in terms of its redshift, often known as z. To calculate the z redshift of an object, you look for an emission or absorption line you can identify, such as those of hydrogen. You then compare the observed wavelength of the line from the object with the standard (not redshifted) line. The difference between the observed and standard wavelengths divided by the standard gives you a number known as z.
If there is no redshift, then there is no difference between observed and standard lines, hence the z is zero. Redshift is thus given by a positive number, where the bigger the number the bigger the z. Technically there is no limit to the value z can have, but the highest we have observed is about z = 12. Bigger z also means greater distance. Because of the expansion of the universe, the light of a distant galaxy is redshifted more than the light of a closer galaxy. So the galaxy with the greatest redshift is the most distant.
The reason astronomers usually talk about redshift instead of distance is that the measured z is purely an observational result. Yes, we know that bigger z means greater distance, but the exact distance depends on the model you use for the universe. We know this model is relatively accurate for determining distances, but with z you don’t have to assume any model.
Since this particular galaxy has a z of 7.51, just how far away is it? The first thing you need to do is transform the redshift to the travel time of the light since it left the galaxy. Once the light left the galaxy, cosmic expansion meant the light redshifted while it travelled. Using the standard model of cosmology we can determine the light left z8 GND 5296 about 13.1 billion years ago.
You might think calculating the distance from that age is simple. After all the speed of light is a constant, and if it travelled 13.1 billion years it must be 13.1 billion light years away. But the universe has been expanding throughout its history, so that answer doesn’t work. We can’t even say that the galaxy was 13.1 billion light years away when the light left because the universe expanded while the light travelled. So the galaxy was actually closer than that when the light left.
The distance the galaxy was from us when the light began its journey can be calculated by what is known as the angular diameter distance. For this galaxy that turns out to be about 3.4 billion light years. That means the light from z8 GND 5296 began its journey 3.4 billion light years away, but due to the expansion of the universe took 13.1 billion years to reach us.
To calculate the distance of the galaxy now, you need to start with the fact that it was 3.4 billion light years away from us 13.1 billion years ago and calculate how much the universe has expanded since then. This is known as the comoving distance. This comes out to be about 29.3 billion light years.
So the light from z8 GND 5296 has a redshift of z = 7.51. That means the light left the galaxy 13.1 billion years ago when the galaxy was 3.4 billion light years away. It is now 29.3 billion light years away. That can be a bit hard to wrap your head around.
Which is yet another reason why astronomers focus on z.